Resolvents of operators and partial functional differential equations with nonautonomous past

To a backward evolution family on a Banach space X we associate an abstract differential operator G through the integral equation on a Banach space of X-valued functions on . We compute the resolvent of the restriction of this operator to a smaller domain to obtain a generator. We then apply the results to prove existence, exponential stability and exponential dichotomy of solutions to partial functional equations with nonautonomous past as discussed in [S. Brendle, R. Nagel, Dist. Contin. Dynam. Systems 8 (2002) 953-966]. Our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria.     

序半群的阿基米德半群的半格分解

本文通过一个序半群S上的一些二元关系以及它的理想的根集的性质该序半群是阿基米德半群的半格,特别地是阿基米德半群的链的刻划,证明了S是阿基米德链当且仅当S是准素的.通过序半群的m-系的概念,证明了S的任意半素理想是含它的所有素理想的交,并通过该结论,证明了S是阿基米德半群的链当且仅当S是阿基米德半群的半格且S的所有素理想关于集包含关系构成链.作为应用,该结论在一般的半群（没有序）[1]中也成立.     

α-次积分半群的Trotter-kato定理

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＷｈｅｎｗｅｉｎｔｅｇｒａｔｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ，ｗｅｃａｎｆｏｒｍｕｌａｔｅｍａｎｙｏｆｔｈｅｍａｓａｎａｂｓｔｒａｃｔＣａｕｃｈｙｐｒｏｂｌｅｍ．ＡｎａｔｕｒａｌｑｕｅｓｔｉｏｎｉｎｔｈｅｔｈｅｏｒｙｉｓｔｏｆｉｎｄｓｕｉｔａｂｌｅｃｏｎｄｉｔｉｏｎｓｕｎｄｅｒｗｈｉｃｈａｓｅｑｕｅｎｃｅｏｆａａｂｓｔｒａｃｔＣａｕｃｈｙｐｒｏｂｌｅｍ．ｕ′ｎ＝Ａｎｕｎ（ｔ），ｕｎ（０）＝ｘｎ，ｆｏｒｔ≥０，ｎ＝１，２，…ｉｎａｎａｒｂｔｉｒａｒｙＢａ…     

Smoothing properties of transition semigroups in hilbert spaces

The paper gives conditions under which the transition semigroup corresponding to a large class of semilinear equations on a Hilbert space transforms Borel functions onto Frechet differentia hies ones.     

On supercyclicity of operators from a supercyclic semigroup

We show that for every supercyclic strongly continuous operator semigroup {Tt}_t?0 acting on a complex F-space, every Tt with t>0 is supercyclic. Moreover, the set of supercyclic vectors of each Tt with t>0 is exactly the set of supercyclic vectors of the entire semigroup.     

One-parameter Lie groups admitted by time-dependent Schrödinger equation: Atoms,molecules and nucleons in harmonic oscillator field

Lie’s method of differential equation is used to obtain the one-parameter Lie groups admitted by the time-dependent Schrödinger equations for atoms, molecules and nucleons in harmonic oscillator field. This group for atoms and molecules is isomorphic to 10-parameter inhomogeneous orthogonal group in 4 dimensions, irrespective of the numbers of nuclei and electrons. For Z protons andN neutrons in a harmonic oscillator field, both isotropic and anisotropic, the r-parameter Lie groups are seraidirect products of an invariant subgroup and a factor group. In the case of isotropic oscillator field r is 1/2[3Z(3Z-1) +3N (3N -1)+2], while for the anisotropic oscillator field r is 1/2[3Z (Z+1)+3N(JV+1)+2].     

FINITELY GENERATED COMMUTATIVE ARCHIMEDEAN SEMIGROUPS

In this work commutative Archimedean finitely generated semigroups are characterized in terms of ideal extensions.     

Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows

In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy.     

Nonlocality of the Misra-Prigogine-Courbage semigroup

We show that the Markov semigroups constructed by Misra, Prigogine, and Courbage through nonunitary similarity transformations of Kolmogorov systems are not implementable by local point transformations, i.e., they are not the Frobenius-Perron semigroups associated with noninvertible point transformations, in contrast with the semigroups obtained by coarse-graining projections. Our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by Goldstein, Misra, and Courbage and also of the previous illustration by Misra and Prigogine for the baker transformation and completes the characterization of the Misra-Prigogine-Courbage semigroups.     

夹心半群S(X,Y,θ)上的α-同余

本文讨论了夹心半群T(X,Y,θ)上的α-同余与集合Y上Tθ-等价关系之间的联系,证明了对于每个Tθ-等价关系E,夹心半群同余格C(T(X,Y,θ))的完全子格γ-1(E)中的最大元是一个α-同余,并判明Symons同余l,d都是α-同余.对于某些拓扑空间X,Y,确定了夹心半群S(X,Y,θ)上的最小(最大)真α-同余.